Question

Find the values of a and b for which x = 3 / 4 and x = - 2 are the roots of the equation ax^2 + bx - 6 = 0.

Answer 1

given
X=3/4 and x=-2 are roots of the equation ax²+b-6
hence
ax²+bx-6=(4x-3)(x+2)
ax²+bx-6=4x²+5x-6
comparing both sides
we get
a=4
and b=5

Answer 2

ax² + bx - 6 = 0

When x = 3/4,

a(3/4)² + b(3/4) - 6 = 0

9/16 a + 3/4 b  - 6= 0

9a + 12b - 96 = 0 ------------------------- [ 1 ]

When x = -2,

a(-2)² + b(-2) - 6 = 0

4a - 2b - 6 = 0

24a - 12b - 36 = 0 ------------------------- [ 2 ]

Find the value of a:

[ 1 ] + [ 2 ]:

33a - 132 = 0

33a = 132

a = 4

Find the value of b:

9a + 12b - 96 = 0

9(4) + 12b - 96 = 0

36 + 12b - 96 = 0

12b = 60

b = 5

Answer: a = 4 , b = 5